A strategy profile $(x, y) \in X \times Y$ is a Nash equilibrium of the zero-sum game with payoff matrix $P$ if and only if $x$ and $y$ are optimal solutions to the maximin and minimax programs respectively, i.e. \begin{align*} \min_{y' \in Y} p(x, y') &= \max_{x' \in X} \min_{y' \in Y} p(x', y'), \\ \max_{x' \in X} p(x', y) &= \min_{y' \in Y} \max_{x' \in X} p(x', y'). \end{align*}